JP7613170B2 - How to create a filler model - Google Patents

Description

The present invention relates to a method for creating a filler model.

In recent years, various simulation methods (numerical calculations) have been proposed to evaluate the properties of composite materials consisting of fillers and polymeric materials using computers in order to develop polymeric materials such as rubber compounds. In this type of simulation method, a filler model is created that models the filler (see, for example, Patent Document 1 below).

Patent No. 5913260

In the method of Patent Document 1, multiple filler particle models are arranged in a crystal lattice pattern, such as a face-centered cubic lattice pattern, based on a three-dimensional image of a polymeric material, and are then bonded together to create a filler model. This causes the outer surface of the filler model to be formed in a regular stepped pattern, which creates a problem in that behavior in the simulation does not occur with the original filler. To solve this problem, it is possible to arrange the filler particle models that make up the outer surface of the filler model in an irregular manner, but there is a problem in that creating such a filler model requires a long calculation time.

The present invention was devised in consideration of the above-mentioned circumstances, and its main objective is to provide a method for quickly creating a filler model that can properly represent the outer surface of the filler.

The present invention is a method for creating a filler model for numerical analysis of a filler, comprising the steps of inputting a filler region, which is a region in which the filler exists, into a computer, and inputting an amorphous structure model in which a plurality of particle models are irregularly arranged into the computer, and the steps of the computer defining a virtual space for filler that includes the entire filler region, arranging the plurality of amorphous structure models inside the virtual space for filler, and deleting the particle models that exist outside the filler region to create the filler model.

The filler model creation method according to the present invention may further include, after the step of creating the filler model, a step in which the computer modifies the filler model so that the filling rate of the particle model inside the filler model is smaller than the filling rate of the particle model in the outer layer of the filler model.

In the method for creating the filler model according to the present invention, the step of modifying the filler model may include a step of arranging the particle model inside the filler model based on a crystal structure having a smaller filling rate than the outer layer of the filler model.

In the filler model creation method according to the present invention, the step of creating the filler model may include a step of deleting the particle models that extend outside the filler region, and then combining adjacent particle models.

The filler model creation method of the present invention, by having the above-mentioned configuration, makes it possible to create a filler model that properly represents the outer surface of the filler in a short period of time.

FIG. 1 is a perspective view showing an example of a computer used in a method for creating a filler model. 13 is a flowchart showing an example of a processing procedure of a filler model creating method. 13A and 13B are conceptual diagrams showing an example of a filler region and a filler virtual space. FIG. 2 is a conceptual diagram showing an example of an amorphous structure model. FIG. 13 is a conceptual diagram showing an example of a virtual space for a filler in which a plurality of amorphous structure models are arranged. FIG. 1 is a conceptual diagram illustrating an example of a filler model. 13 is a flowchart showing an example of a processing procedure of a filler model creation method according to another embodiment of the present invention. FIG. 11 is a cross-sectional view showing an example of a filler model according to another embodiment of the present invention. This is the structural formula of polybutadiene. 1 is a flowchart showing an example of a processing procedure of a simulation method for a polymer material. FIG. 2 is a conceptual diagram showing an example of a polymer material model. FIG. 1 is a conceptual diagram showing an example of a filler model whose outer surface is formed in a regular stepped pattern.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
In the filler model creation method of this embodiment (hereinafter, sometimes simply referred to as the "creation method"), a model for numerical analysis of the filler (filler model) is created. The filler model created by this creation method is used in the polymer material simulation method described below (hereinafter, sometimes simply referred to as the "simulation method"). In this embodiment, a computer is used in the creation method and the simulation method.

Figure 1 is a perspective view showing an example of a computer 1 used in the filler model creation method. The computer 1 includes a main body 1a, a keyboard 1b, a mouse 1c, and a display device 1d. The main body 1a is provided with a central processing unit (CPU), a ROM, a working memory, a storage device such as a magnetic disk, and disk drive devices 1a1 and 1a2. The storage device stores in advance the processing procedures (programs) for executing the creation method and simulation method of this embodiment.

The filler is, for example, silica, carbon black, alumina, etc. In this embodiment, the filler is, for example, spherical. The filler is dispersed in the polymer material, for example, as an aggregate in which multiple spherical primary particles are bonded together, or as an agglomerate in which these particles are aggregated.

Figure 2 is a flowchart showing an example of the processing procedure of the filler model creation method. In the creation method of this embodiment, first, a filler region 17, which is a region where filler exists, is input to the computer 1 (step S1). The filler region 17 corresponds to a region in which a particle model 11, described below, can be placed in the filler model 10 created by the creation method of this embodiment. Figures 3(a) and (b) are conceptual diagrams showing an example of the filler region 17 and the virtual space 6 for filler.

The shape of the filler region 17 can be set as appropriate. The shape of the filler region 17 may be set based on the actual three-dimensional structure of the filler identified by 3DTEM observation, for example. In step S1, for example, from a collection of partial regions obtained by dividing the three-dimensional structure so that the length, width, and height are equally spaced, all sets of partial regions in which the filler exists may be selected, and the entire set may be input as the filler region 17. Note that the filler region 17 does not need to be a single continuous mass, and may include multiple isolated aggregates or higher-order structures such as aggregates.

As shown in FIG. 3, in this embodiment, the filler region 17 is represented as an arrangement of one or more primary particles (primary particle model 5 modeling the primary particles) having a certain particle size. As shown in FIG. 3(a), for example, when the filler model 10 is composed of one primary particle model 5, the filler region 17 is defined as one spherical region. On the other hand, as shown in FIG. 3(b), for example, when the filler model 10 is composed of multiple primary particle models 5, the filler region 17 is defined as multiple isolated aggregates composed of multiple primary particles bonded together.

The filler region 17 in this embodiment is input as a pair of the particle diameter D1 of the primary particle model 5 shown in FIG. 3 and the center of gravity position of each primary particle model 5. At this time, the filler region 17 is defined as a region that includes, for all primary particle models 5, a spherical region in which the distance from the center of gravity of each primary particle model 5 is half or less of the particle diameter D1. Note that the filler region 17 may include overlapping spherical regions of multiple primary particle models 5. In such a case, the primary particle models 5 are considered to be aggregates in which the primary particle models 5 are bonded to each other. Note that if the overlap is too small, after the particle models 11 are placed in the filler region 17, a problem may occur in which the particle models 11, 11 are not adjacent to each other (are spaced apart) between the primary particle models 5, 5 that are bonded to each other. In order to avoid such a problem, it is desirable that the size (thickness) of the overlap between the bonded primary particle models 5, 5 is sufficiently larger than the diameter of the particle model 11.

The filler model 10 of this embodiment is exemplified as being mainly composed of one primary particle model 5 shown in FIG. 3(a). Therefore, in step S1, a filler region 17 consisting of one primary particle model 5 is defined inside the filler virtual space 6. In the filler region 17 of this embodiment, the particle diameter D1 of the spherical primary particle model 5 and the position of the center of gravity of the primary particle model 5 are defined. The filler region 17 is stored in the computer 1.

Next, in the creation method of this embodiment, an amorphous structure model 22 is input to the computer 1 (step S2). The amorphous structure model 22 is a virtual space 19 for amorphous structure described below in which a plurality of particle models 11 are irregularly arranged (filled). FIG. 4 is a conceptual diagram showing an example of the amorphous structure model 22.

In step S2 of this embodiment, first, the diameter D2 of the particle model (coarse-grained particle model) 11 to be placed inside the amorphous structure model 22 is input to the computer 1. The particle model 11 can be appropriately defined based on this diameter D2.

The particle model 11 in this embodiment is a coarse-grained group of atoms that make up the filler. The particle model 11 is treated as a mass point in the equation of motion in the molecular dynamics calculation described below. That is, parameters such as mass, volume, diameter D2, charge, and initial coordinates are defined for the particle model 11. The particle model 11 (diameter D2) is input to the computer 1.

Next, in step S2 of this embodiment, the amorphous virtual space 19 is input to the computer 1. The amorphous virtual space 19 has, for example, periodic boundary conditions defined, and a three-dimensional shape that can be filled in the space without gaps is appropriately defined.

In this embodiment, the amorphous virtual space 19 is defined as a cube having three pairs of amorphous planes 20, 20 that face each other. A periodic boundary condition is defined for each amorphous plane 20. Therefore, the amorphous virtual space 19 is treated as one amorphous plane 20 being continuous (connected) with the amorphous plane 20 on the opposite side.

The length of one side of the amorphous virtual space 19 (amorphous length) Lc can be set as appropriate. Note that the larger the amorphous length Lc, the more particle models 11 are placed in the amorphous virtual space 19, and the longer it takes to create the amorphous structure model 22. For this reason, it is desirable to set the amorphous length Lc to a sufficiently small value that is less than half the size of the filler length La (shown in Figures 3(a) and (b)). This makes it possible to shorten the time required to create the amorphous structure model 22 input in step S2, and thus shorten the time required to create the filler model 10.

On the other hand, the shorter the amorphous length Lc, the smaller the number of particle models 11 that can be arranged. In particular, if the amorphous length Lc is extremely short, it becomes difficult to arrange the particle models 11 irregularly. In this case, if such amorphous structure models 22 are arranged repeatedly (in large numbers) in step S4 (shown in FIG. 2) of arranging the amorphous structure models described below, the outer surface of the filler model 10 (shown in FIG. 6) may become more regular. From this perspective, it is desirable to set the amorphous length Lc to a sufficiently large value so that the number of particle models 11 is 50 or more at a predetermined filling rate of the amorphous structure (for example, 60% to 68%).

By setting the amorphous length Lc to such a value, it is possible to prevent the amorphous length Lc from becoming smaller than necessary. This makes it easy to create an amorphous structure model 22 in which particle models 11 are irregularly arranged inside the amorphous virtual space 19 in step S2. Furthermore, the effect of the outer surface of the filler model 10 (shown in FIG. 6) becoming regular due to the repeated arrangement of the amorphous structure models 22 can be sufficiently reduced.

In addition, when a periodic boundary condition is defined on the filler plane 7 (shown in FIG. 3(a)), in step S4 (shown in FIG. 2) of arranging the amorphous structure model 22 described below, it is desirable to set the amorphous length Lc so that the size of the assembly (filler model 10) in which the amorphous structure models 22 are arranged one-dimensionally an integer number of times is sufficiently close to the filler length La. For this reason, it is desirable to set the amorphous length Lc to a value obtained by dividing the filler length La by an integer of 2 or more.

Next, in step S2 of this embodiment, multiple particle models 11 are placed in the amorphous virtual space 19. The number of particles in the particle model 11 is defined appropriately. The number of particles in the particle model 11 may be selected so as to be the filling rate of the amorphous structure (e.g., 60% to 68%), for example.

The particle models 11 are appropriately arranged. In this embodiment, first, multiple particle models 11 are randomly arranged in the amorphous virtual space 19, allowing overlap between the particle models 11, 11. Next, in this embodiment, a constant volume molecular dynamics calculation is performed using the force field of formula 2.4 described in Paper 1 (Kurt Kremer & Gary S. Grest, "Dynamics of entangled linear polymer melts: A molecular-dynamics simulation", J. Chem Phys. vol.92, No.8, 15 April 1990), and then the overlap between the particle models (coarse-grained particle models) 11, 11 is removed.

Next, in step S2 of this embodiment, a constant volume molecular dynamics calculation is performed using the force field of formula A1 described in paper 1. Then, a liquid state of the particle model (coarse-grained particle model) 11 is created, and the same force field is used to perform a potential energy minimization calculation, so that the arrangement of the particle model 11 can be created. Note that instead of using the above-mentioned method, the arrangement of the particle model 11 may be created by performing a calculation using the molecular dynamics method of hard spheres or the Monte Carlo method.

Molecular mechanics calculations (molecular dynamics calculations) can be performed, for example, using COGNAC, which is included in the Soft Material Integrated Simulator (J-OCTA) manufactured by JSOL Co., Ltd.

In step S2 of this embodiment, the amorphous structure model 22 is input by randomly arranging multiple particle models 11 in the amorphous virtual space 19 based on the above-mentioned method. The amorphous virtual space 19 and the amorphous structure model 22 are stored in the computer 1.

Next, in the creation method of this embodiment, a filler virtual space (virtual space) 6 for creating a filler model is input to the computer 1 (step S3). As shown in FIG. 3, the filler virtual space 6 of this embodiment is defined so as to include the entire filler region 17. The shape and size of the filler virtual space 6 can be defined as appropriate as long as it includes the entire filler region 17.

The filler virtual space 6 in this embodiment may be defined as a rectangular parallelepiped region having a length obtained by adding the particle diameter D1 of two primary particle models 5 to the length of the filler region 17 in the x-axis direction, the length of the filler region 17 in the y-axis direction, and the length of the filler region 17 in the z-axis direction (i.e., the upper and lower limits of the x-, y-, and z-coordinates of the filler region 17 are each expanded by the width of the particle diameter D1 of the primary particle model 5). In addition, when a periodic boundary condition is imposed on the coordinate system in which the primary particle model 5 of the filler region 17 is arranged, the entire coordinate system to which the periodic boundary condition is imposed may be defined as the filler virtual space 6. At this time, the periodic boundary condition required to define the filler virtual space 6 may be input together with the input of the filler region 17.

In this embodiment, the virtual space 6 for the filler is exemplified as being defined as a cube having three pairs of planes 7, 7 that face each other. A periodic boundary condition is defined for each plane 7. Therefore, the virtual space 6 for the filler is treated as one plane 7 being continuous (connected) with the opposite plane 7.

The length of one side of the virtual space for filler 6 (length for filler) La can be set as appropriate. In this embodiment, the length for filler La is desirably set so that the volume is as small as possible while including the entire filler region 17. This allows the particle model 11 described below to be filled into the filler region 17 without gaps, and further reduces the time required to place the amorphous structure model 22 (shown in FIG. 4) in step S4 described below. The virtual space for filler 6 is stored in the computer 1.

Next, in the creation method of this embodiment, the computer 1 places multiple amorphous structure models 22 inside the filler virtual space 6 (step S4). In step S4, the multiple amorphous structure models 22 are placed so that the particle models 11 are filled in the filler region 17. FIG. 5 is a conceptual diagram showing an example of the filler virtual space 6 in which multiple amorphous structure models 22 are placed.

In step S4 of this embodiment, first, the number of times N to repeatedly place the amorphous structure model 22 in each of the x-axis direction, y-axis direction, and z-axis direction in the filler virtual space 6 is determined. The number of times N can be determined appropriately once the particle model 11 is filled in the filler region 17. The number of times N is obtained, for example, by rounding off (rounding off the decimal point) the value obtained by dividing the filler length La by the amorphous length Lc. In order to place the amorphous structure model 22 without dividing it, it is desirable that the number of times N is an integer of 2 or more. Note that if the obtained number of times N is not an integer of 2 or more, the values of the amorphous length Lc and the like may be changed, and the steps S2 to S4 may be performed again. The number of times N is stored in the computer 1.

Next, in step S4 of this embodiment, the computer 1 performs the following in the filler virtual space 6:
The amorphous structure models 22 are repeatedly arranged in each of the x-axis direction, the y-axis direction, and the z-axis direction. In this embodiment, the arrangement of the amorphous structure models 22 is determined based on the determined number of times N (in FIG. 5, the number of times N=3). Within the filler virtual space 6, a three-dimensional vector R _i (i _x , i _y , i _z ) indicating the coordinates of each particle model 11 included in each amorphous structure model 22 is defined by the following formula (1).

ここで、式（１）の変数は、以下のとおりである。
Ｌ_ａ：フィラー用長さ
Ｌ_ｃ：アモルファス用長さ
Ｎ：回数
ｒ_i：各アモルファス構造モデル２２に含まれるｉ番目の粒子モデル１１について、アモルファス用仮想空間１９内での座標を示す３次元ベクトル
ｅ_x：ｘ方向の単位ベクトル（１，０，０）
ｅ_y：ｙ方向の単位ベクトル（０，１，０）
ｅ_z：ｚ方向の単位ベクトル（０，０，１）

Here, the variables in equation (1) are as follows:
L _a : Length for filler L _c : Length for amorphous N: Number of times r _i : Three-dimensional vector indicating the coordinates in the amorphous virtual space 19 for the i-th particle model 11 included in each amorphous structure model 22 e _x : Unit vector (1, 0, 0) in the x direction
e _y : unit vector in the y direction (0, 1, 0)
e _z : unit vector in the z direction (0,0,1)

In the above formula (1), R _i (i _x , i _y , i _z ) indicates the three-dimensional vector of the coordinates of the i-th particle model 11 of the amorphous structure model 22 arranged at the ixth position in the x-axis direction, the iyth position in the y-axis direction, and the izth position in the z-axis direction in the virtual space 6 for the filler.

In this embodiment, the coordinates of the particle model 11 are determined based on the three-dimensional vector R _i (i _x , i _y , i _z ) in the above formula (1), and thus the arrangement of the amorphous structure models 22 repeatedly arranged inside the virtual space 6 for the filler (in the x-axis direction, y-axis direction, and z-axis direction) can be determined.

According to the above formula (1), the amorphous structure model 22 is enlarged or reduced in each of the x-axis direction, y-axis direction, and z-axis direction based on a magnification equal to the value obtained by dividing the filler length La by the product of the number of times N and the amorphous length Lc. This corrects the "deviation" caused by the rounding performed when the number of times N is calculated, so that the amorphous structure model 22 is arranged almost without gaps in the filler virtual space 6.

In the creation method of this embodiment, the amorphous structure model 22 of the particle model 11 is input in the amorphous virtual space 19, which is smaller than the filler virtual space 6 shown in FIG. 4. Therefore, in this embodiment, the number of particles of the particle model 11 for which the amorphous structure model 22 is created can be reduced compared to when the amorphous structure model (not shown) of the particle model 11 is calculated in the filler virtual space 6, which is larger than the amorphous virtual space 19. Note that the calculation of repeatedly arranging the amorphous structure model 22 as in this embodiment is so small that the calculation time can be ignored compared to molecular dynamics calculations, etc. Therefore, in step S4 of this embodiment, the calculation time can be significantly reduced. The filler virtual space 6 in which multiple amorphous structure models 22 are arranged is stored in the computer 1.

Next, in the creation method of this embodiment, the computer 1 deletes the particle model 11 that protrudes from the filler region 17 (shown by the two-dot chain line in FIG. 5) to create the filler model 10 (shown in FIG. 6) (step S5). The determination of whether the particle model 11 protrudes from the filler region 17 can be defined as appropriate. In this embodiment, if a part of the particle model 11 exists outside the filler region 17, it is determined that the particle model 11 protrudes from the filler region 17. As a specific example of such a determination, the particle model 11 may be determined to protrude when the center coordinates of the particle model 11 are outside the filler region 17, or when the shortest distance between the center coordinates of the particle model 11 and the outer surface of the filler region 17 is smaller than the diameter D2 (shown in FIG. 4) of the particle model 11 divided by 2.

In step S5 of this embodiment, the particle models 11 that protrude from the filler region 17 are deleted, and a filler model 10 in which the primary particle model 5 is defined is created based on the particle models 11 filled in the filler region 17. FIG. 6 is a conceptual diagram showing an example of the filler model 10. The filler model 10 is stored in the computer 1.

Next, in the creation method of this embodiment, the computer 1 combines adjacent particle models 11, 11 (step S6). Step S6 in this embodiment is performed after deleting the particle models 11 that protrude from the filler region 17 (shown in FIG. 5).

In step S6 of this embodiment, adjacent particle models 11, 11 that are separated by a predetermined distance (hereinafter, sometimes simply referred to as the "bonding distance") or less are bonded. The bond distance can be set appropriately. In this embodiment, the bond distance can be set, for example, in the range of 1.0 to 1.5 times (1.1 times in this example) the diameter D2 (shown in FIG. 4) of the particle model 11.

In step S6 of this embodiment, as shown in FIG. 6, adjacent particle models 11, 11 that are equal to or less than the bond distance are bonded by a bond chain model 8. As a result, in step S6, a filler model 10 is created in which multiple particle models 11 are bonded. The rigidity of such a filler model 10 can be increased by the bond chain model 8. The filler model 10 is stored in the computer 1.

In the creation process of this embodiment, the filler model 10 in which the particle models 11, 11 are bonded is created using the bond chain model 8, but the invention is not limited to this embodiment. For example, step S6 may be omitted to create a filler model 10 in which the particle models 11, 11 are not bonded. In this case, the coordinates of all the particle models 11 constituting the filler model 10 may be fixed, or a constraint condition may be defined, such as making each aggregate a rigid body.

In the creation method of this embodiment, unlike the filler particle model of Patent Document 1 (i.e., the coarse-grained particle model of this embodiment) arranged in a crystal lattice pattern, the particle model 11 of the filler model 10 is arranged in an irregular amorphous (non-crystalline) shape. As a result, in the creation method of this embodiment, the outer surface of the filler model 10 becomes irregular as shown in FIG. 6, and it is possible to prevent it from being formed in a regular step shape as shown in FIG. 12. Note that, in the case of silica produced by, for example, a sintering method or a precipitation method, the outer surface of an actual filler is not formed in a regular step shape as shown in FIG. 12. Therefore, in the creation method of this embodiment, it is possible to create a filler model 10 that appropriately represents the outer surface of an actual filler.

Next, in the creation method of this embodiment, in order to use the created filler model 10 in a molecular dynamics calculation, a force field (not shown) for molecular dynamics calculation may be set between the particle models 11, 11 (step S7). As the force field, for example, an LJ potential in which the repulsive force between the particle models 11, 11 is expressed by the following formula (2) described in Paper 1 and a cutoff distance r _c = 2 ^1/6 σ, and a bond potential in which the bond between the particle models 11, 11 is expressed by the following formula (3) may be used.

ここで、式（２）の変数は、次のとおりである。
ｒ：粒子モデル１１、１１の中心間の距離
ε：ポテンシャルエネルギーの大きさに関するパラメータ
σ：粗視化シミュレーションに用いられる大きさに関するパラメータ
ｒ_c：カットオフ距離ｒ_c

where the variables in equation (2) are as follows:
r: distance between the centers of the particle models 11, 11 ε: parameter related to the magnitude of the potential energy σ: parameter related to the magnitude used in the coarse-grained simulation r _c : cutoff distance r _c

ここで、式（３）の変数は、次のとおりである。
ｒ：粒子モデル１１、１１の中心間の距離
ｋ：ポテンシャルエネルギーの大きさに関するパラメータ
Ｒ₀：伸びきり長

where the variables in equation (3) are as follows:
r: distance between the centers of the particle models 11, 11 k: parameter related to the magnitude of potential energy R ₀ : stretched length

In this embodiment, the distance between the adjacent particle models 11, 11 of the filler model 10 is created so that both are close to the diameter D2 (shown in FIG. 4). The diameter D2 is set equal to the cutoff distance r _c = 2 ^1/6 σ = about 1.122σ. For this reason, if the bond potential shown in the above formula (3) is used as is in the bond chain model 8, the bond length before the bond potential of the above formula (3) is used is about 1.122σ, while the equilibrium length of the bond potential is about 0.961σ, so that relaxation occurs due to molecular dynamics calculation, and the entire filler model may shrink. In order to avoid this problem, the bond potential between the particle models 11, 11 may be adjusted so that the equilibrium length of the bond potential is equal to the cutoff distance r _c . For example, the parameter σ (term of U _LJ (r)) and the stretched length R ₀ of the bond potential between the particle models 11, 11 may both be multiplied by 1.168.

In the simulation method described below, molecular dynamics calculations are performed on all particle models 11 constituting the filler model 10 shown in FIG. 6. For this reason, if the filling rate (number) of the particle models 11 is large, the calculation time tends to increase. Therefore, in the creation method, the filler model 10 may be modified so that the filling rate of the particle models 11 is small. FIG. 7 is a flowchart showing an example of the processing procedure of the creation method of a filler model according to another embodiment of the present invention. FIG. 8 is a cross-sectional view showing an example of a filler model 10 according to another embodiment of the present invention. In FIG. 8, the virtual space 6 for filler is omitted. In this embodiment, the same components as those in the previous embodiments are given the same reference numerals, and the description may be omitted.

In the creation method of this embodiment, the filler model 10 is modified so that the filling rate of the particle model 11 inside the filler model 10 is smaller than the filling rate of the particle model 11 in the outer layer 21 of the filler model 10. This modification process includes a step S8 of making the filler model 10 hollow, and a step S9 of filling the hollow filler model 10 with particle models 11 of a crystalline structure, which has a filling rate lower than that of an amorphous structure.

In step S8, the particle models 11 inside the outer layer 21 of the filler model 10 are deleted while maintaining the particle models 11 of the outer layer 21. In this embodiment, all particle models 11 in the filler virtual space 6 that are not included in the outer layer 21 are deleted.

The outer layer 21 can be defined as appropriate. In this embodiment, the outer layer 21 is defined as a region within a predetermined thickness r1 from the outer surface of the filler region 17 toward the inside. The thickness r1 can be defined as appropriate. In this embodiment, the thickness r1 is set to 5% to 20% (10% in this example) of the particle diameter D1 (shown in FIG. 3(a)) of the primary particle model 5.

The determination of whether or not the particle model 11 is included in the outer layer 21 can be made as appropriate. In this embodiment, if at least a part of the particle model 11 exists outside the outer layer 21, it is determined that the particle model 11 is not included in the outer layer 21. More specifically, if the center coordinates of the particle model 11 exist outside the outer layer 21, or if the shortest distance between the center coordinates of the particle model 11 and the outer surface of the outer layer 21 is smaller than the diameter D2 (shown in FIG. 4) of the particle model 11 divided by 2, it is determined that the particle model 11 is not included in the outer layer 21.

In step S9, the particle model 11 having a crystalline structure with a filling rate lower (smaller) than that of the amorphous structure is filled into the internal region of the outer layer 21 of the filler model 10. Any crystalline structure can be used as appropriate, as long as it is smaller than the filling rate of the outer layer 21 (e.g., 60% to 68%). In this embodiment, a diamond structure (filling rate of 34%) is used.

In the creation method of this embodiment, the amorphous arrangement of the particle models 11 that make up the outer layer 21 is maintained, so that, as in the previous embodiments, a filler model 10 that can more appropriately represent the outer surface of the filler can be created. Furthermore, in the creation method of this embodiment, the filling rate of the particle models 11 inside the filler model 10 can be reduced while suppressing a decrease in the strength of the entire filler model 10, so that the number of particle models 11 to be calculated in the simulation method described below can be reduced. Therefore, the creation method of this embodiment makes it possible to shorten the calculation time while maintaining the calculation accuracy in the simulation method (molecular dynamics calculation) described below.

In this embodiment, for the crystal structure filled inside the filler model 10, adjacent particle models 11 may be connected by a bond chain model 8 (shown in FIG. 6). Furthermore, the particle model 11 constituting the crystal structure and the particle model 11 constituting the outer layer 21 may be connected by a bond chain model 8. This makes it possible to reduce the number of particle models 11 inside the filler model 10 while suppressing a decrease in the strength of the filler model 10.

Next, a method for simulating a polymeric material using the filler model 10 obtained by the above-mentioned creation method will be described. In the simulation method of this embodiment, the performance of a polymeric material containing a filler and a polymer component is calculated using a computer 1.

The polymer component includes, for example, natural rubber, synthetic rubber, and resin. An example of the polymer component of this embodiment is cis-1,4 polybutadiene (hereinafter, sometimes simply referred to as "polybutadiene"). Figure 9 shows the structural formula of polybutadiene.

The molecular chains 12 constituting polybutadiene are formed by linking monomers 13 {-[CH ₂ -CH=CH-CH ₂ ]-} consisting of a methylene group (-CH ₂ -) and a methine group (-CH-) with a degree of polymerization n. Moreover, a methyl group (-CH ₃ ) is linked to the end of the polymer material instead of the methylene group (-CH ₂ ). Note that the polymer component may be something other than polybutadiene. The filler is as described above. FIG. 10 is a flow chart showing an example of the processing procedure of the simulation method for polymer materials.

In the simulation method of this embodiment, first, a cell, which is a virtual space corresponding to a part of a polymer material (not shown), is input to the computer 1 (step S10). FIG. 11 is a conceptual diagram showing an example of a polymer material model. In FIG. 11, only a filler model 10 and a part of a molecular chain model 15 are shown as representatives.

The cell 14 is a region that is the subject of calculation in the molecular dynamics calculation described below. The cell 14 in this embodiment can be set, for example, in the same manner as the filler virtual space 6 shown in FIG. 3(a). At least one filler model 10 and multiple molecular chain models 15 are placed inside the cell 14.

The length Lb of one side of the cell 14 can be set appropriately. In this embodiment, the length Lb is preferably three times or more the radius of gyration (not shown), which is an amount indicating the spread of the molecular chain model 15 described below. This allows the cell 14 to prevent collisions with itself (image), which is arranged periodically due to the periodic boundary condition, in molecular dynamics calculations, and to properly calculate the spatial spread of the molecular chain model 15. In addition, the size of the cell 14 is set to a stable volume at, for example, 1 atmosphere. This allows the cell 14 to define at least a portion of the volume of the polymer material to be analyzed. The cell 14 is stored in the computer 1.

Next, in the simulation method of this embodiment, a molecular chain model 15 that models a molecular chain 12 (shown in FIG. 9) of a polymeric material is input to the computer 1 (step S11). The molecular chain model 15 of this embodiment is defined as a coarse-grained molecular model.

The molecular chain model 15 is represented using a number of particles 16, the number of which is less than the number of atoms that make up the molecular chain 12 (shown in FIG. 9). These particles 16, 16 are connected by bonding chains (not shown). The molecular chain model 15 of this embodiment is exemplified as a Kremer-Grest model, but is not particularly limited thereto. The molecular chain model 15 may be, for example, a model based on DPD (dissipative particle dynamics).

In this embodiment, each particle 16 corresponds to, for example, a monomer 13 of the molecular chain 12 shown in FIG. 9. When the molecular chain 12 is polybutadiene, for example, 1.10 monomers 13 are used as a structural unit, and the structural unit is replaced with one particle 16. As a result, multiple particles 16 (for example, 10 to 5000) are set in the molecular chain model 15 of this embodiment. The particles 16 are treated as mass points in the equation of motion in the molecular dynamics calculation described below. That is, parameters such as mass, diameter, charge, and initial coordinates are defined for the particles 16. Each of these parameters is stored in the computer 1 as numerical information.

The bond chain (not shown) is defined as a bond potential that defines an equilibrium length between particles 16, 16. Here, the "equilibrium length" is defined as the bond distance between particles 16, 16 at which the value of the bond potential is smallest. The bond potential of the bond chain (not shown) can be set, for example, based on the above-mentioned paper 1. Such a molecular chain model 15 is numerical data for handling polymeric materials in molecular dynamics calculations, and is input to computer 1.

Next, in the simulation method of this embodiment, the computer 1 acquires a filler model 10 (step S12). In step S12, the processing procedure of the above-mentioned creation method is performed, and at least one filler model 10 is acquired. The filler model 10 is stored in the computer 1.

Next, in the simulation method of this embodiment, the computer 1 places the molecular chain model 15 and the filler model 10 inside the cell 14 (step S13). In step S13 of this embodiment, first, the filler model 10 is placed inside the cell 14. The position at which the filler model is placed can be set appropriately. For example, the filler model 10 may be placed in an area where the filler is placed in an electron beam transmission image of the polymer material. In this way, in step S13, the filler model 10 can be placed inside the cell 14 based on the relative position of the filler in the actual polymer material.

Next, in step S13, multiple molecular chain models 15 are placed in areas where no filler models 10 are placed. This allows the molecular chain models 15 to be placed in the cells 14 in a manner that approximates the distribution of the actual molecular chains 12.

Next, in the simulation method of this embodiment, a potential P3 is defined in which attractive and repulsive forces act between the particles 16, 16 of adjacent molecular chain models 15, 15, and between the particle 16 of the molecular chain model 15 and the particle model 11 of the filler model 10 (step S14). As this potential P3, the LJ potential that represents the repulsive force described in the above-mentioned Patent Document 1 is adopted. Each variable of the LJ potential can be set appropriately based on, for example, the above-mentioned Paper 1.

Next, in the simulation method of this embodiment, the computer 1 performs molecular dynamics calculations on the molecular chain model 15 and the filler model 10 (step S15). In the molecular dynamics calculations, for example, Newton's equations of motion are applied to the cell 14 for a predetermined time, assuming that the particle model 11 of the filler model 10 and the particles 16 of the molecular chain model 15 follow classical mechanics. Then, the movements of the particle model 11 and the particles 16 at each time are tracked for each unit time.

In step S15, first, the structural relaxation of the molecular chain model 15 is calculated. In this case, the pressure and temperature are kept constant, or the volume and temperature are kept constant, in the cell 14. By calculating this structural relaxation, in step S15, a polymer material model 18 that models the polymer material (not shown) is created.

Next, in step S15 of this embodiment, a deformation calculation (e.g., elongation deformation) is performed on the created polymer material model 18. Then, the physical quantity (e.g., stress, etc.) obtained by the deformation calculation of the polymer material model 18 is stored in the computer 1.

The simulation method of this embodiment uses a filler model 10 having a surface structure that is similar to the microstructure of the filler, so that the effect of the filler surface structure on the physical properties of the polymer material (molecular chain model 15) can be evaluated with high accuracy. Therefore, the simulation method of this embodiment can accurately calculate the physical quantities obtained by deformation calculation of the polymer material model 18.

Next, in the simulation method of this embodiment, it is determined whether the physical properties of the polymer material model 18 are good or not (step S16). The determination of whether the properties are good or not is appropriately performed based on the strength, durability, etc. required for the polymer material.

If the physical properties are good in step S16 ("Y" in step S16), a polymer material is produced (step S17) based on the structure of the filler and molecular chains 12 (shown in FIG. 9). The produced polymer material is used to produce industrial products such as tires. On the other hand, if the physical properties are not good in step S16 ("N" in step S16), the structure of at least one of the filler and molecular chains 12 is changed (step S18), and steps S10 to S16 are performed again. As a result, the simulation method of this embodiment makes it possible to produce a polymer material with good physical properties and an industrial product using that polymer material.

The above describes in detail a particularly preferred embodiment of the present invention, but the present invention is not limited to the illustrated embodiment and can be modified and implemented in various ways.

Based on the processing procedure shown in FIG. 2, a model for numerical analysis of the filler was created (Example 1). Furthermore, based on the processing procedure shown in FIG. 7, a model for numerical analysis of the filler was created (Example 2). In Example 2, unlike Example 1, a process of modifying the filler model was carried out so that the filling rate of the particle model inside the filler model was smaller than the filling rate of the particle model in the outer layer of the filler model. In Example 2, the crystal structure of the particle model inside the filler model is a diamond structure.

To compare the calculation times of Example 1 and Example 2, a filler model was created by calculating structural relaxation by molecular dynamics calculations for a particle model placed inside the virtual space for the filler without using an amorphous structure model (Comparative Example 1). In Comparative Example 1, for the particle model after structural relaxation, the particle model outside the filler region was removed, and adjacent particle models were bonded together.

Furthermore, in order to compare the filler surface structure, similarly to the above-mentioned Patent Document 1, multiple particle models were arranged in a crystal lattice pattern such as a face-centered cubic lattice pattern, and then the particle model arranged outside the filler region corresponding to one spherical primary particle was deleted to create a filler model. Figure 12 is a conceptual diagram showing an example of the filler model of Comparative Example 2.

Then, the time required to create the filler model was obtained for Example 1, Example 2, Comparative Example 1, and Comparative Example 2. Furthermore, a simulation was performed using the filler models of Example 1, Example 2, Comparative Example 1, and Comparative Example 2 based on the processing procedure shown in Fig. 10, and the time required for the simulation was obtained. The common specifications are as follows.
Filler model (before modification process of Example 2):
Number of primary particle models: 1
Number of particle models: 10,000 Amorphous structure model:
Number of particle models: 80 Length of virtual space for filler La: 32.3σ
Filler volumetric filling rate: 30%
Structural relaxation calculation of amorphous structure model:
Unit time: 0.001 τ
Structural relaxation calculation using the potential shown in equation 2.4 of paper 1
Number of steps: 10,000 steps
Structural relaxation calculation using the LJ potential shown in Equation A1 of Paper 1
Number of steps: 10,000 steps
Potential energy minimization with the above LJ potential
Number of steps: Within 1,000 steps Structural relaxation calculation of polymer material model:
Unit time: 0.006 τ
Number of steps: 1,000,000 steps

As a result of the test, the time required to create the filler models in Examples 1 and 2 was 0.3% of the time required to create the filler model in Comparative Example 1. Therefore, in Examples 1 and 2, the filler models could be created in a short time.

Furthermore, in Example 2, the filling rate of the particle model inside the filler model can be made smaller than the filling rate of the particle model in the outer layer of the filler model, so that the simulation calculation time can be reduced to 89% compared to Example 1 and Comparative Example 1.

In addition, the outer surfaces of the filler models in Examples 1 and 2 were prevented from being formed in a regular stepped pattern as in Comparative Example 2 shown in FIG. 12, and a filler model capable of more appropriately expressing the outer surface of the filler could be created. In the simulations of Examples 1 and 2, the behavior that occurs in the original filler could be reproduced with greater accuracy than in the simulation of Comparative Example 2.

S1: Step of inputting a filler region S2: Step of inputting an amorphous structure model S3: Step of inputting a virtual space for filler S4: Step of arranging a plurality of amorphous structure models inside the virtual space for filler S5: Step of deleting particle models protruding from the filler region

Claims (4)

1. A method for creating a filler model for a numerical analysis of a filler, comprising:
A step of inputting a filler region, which is a region where the filler exists, into a computer;
and inputting an amorphous structure model in which a plurality of particle models are irregularly arranged into the computer;
The computer,
defining a filler virtual space including the entire filler region;
placing a plurality of the amorphous structure models within the virtual space for filler;
and creating the filler model by deleting the particle model that exists outside the filler region.
How to create a filler model. The method for creating a filler model according to claim 1, further comprising, after the step of creating the filler model, a step in which the computer modifies the filler model so that the filling rate of the particle model inside the filler model is smaller than the filling rate of the particle model in the outer layer of the filler model. The method for creating a filler model according to claim 2, wherein the step of modifying the filler model includes a step of arranging the particle model inside the filler model based on a crystal structure having a smaller filling rate than the outer layer of the filler model. The method for creating a filler model according to any one of claims 1 to 3, wherein the step of creating the filler model includes a step of deleting the particle models that extend outside the filler region, and then combining adjacent particle models.

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